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34
Approximation Accuracy, Gradient Methods, and Error Bound for Structured Convex Optimization
, 2009
"... Convex optimization problems arising in applications, possibly as approximations of intractable problems, are often structured and large scale. When the data are noisy, it is of interest to bound the solution error relative to the (unknown) solution of the original noiseless problem. Related to this ..."
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Cited by 38 (1 self)
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Convex optimization problems arising in applications, possibly as approximations of intractable problems, are often structured and large scale. When the data are noisy, it is of interest to bound the solution error relative to the (unknown) solution of the original noiseless problem. Related to this is an error bound for the linear convergence analysis of firstorder gradient methods for solving these problems. Example applications include compressed sensing, variable selection in regression, TVregularized image denoising, and sensor network localization.
Explicit Sensor Network Localization Using Semidefinite Representations and Clique Reductions
 Department of Combinatorics and Optimization, University of Waterloo
, 2009
"... AMS Subject Classification: The sensor network localization, SNL, problem in embedding dimension r, consists of locating the positions of wireless sensors, given only the distances between sensors that are within radio range and the positions of a subset of the sensors (called anchors). Current solu ..."
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Cited by 28 (10 self)
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AMS Subject Classification: The sensor network localization, SNL, problem in embedding dimension r, consists of locating the positions of wireless sensors, given only the distances between sensors that are within radio range and the positions of a subset of the sensors (called anchors). Current solution techniques relax this problem to a weighted, nearest, (positive) semidefinite programming, SDP,completion problem, by using the linear mapping between Euclidean distance matrices, EDM, and semidefinite matrices. The resulting SDP is solved using primaldual interior point solvers, yielding an expensive and inexact solution. This relaxation is highly degenerate in the sense that the feasible set is restricted to a low dimensional face of the SDP cone, implying that the Slater constraint qualification fails. Cliques in the graph of the SNL problem give rise to this degeneracy in the SDP relaxation. In this paper, we take advantage of the absence of the Slater constraint qualification and derive a technique for the SNL problem, with exact data, that explicitly solves the corresponding rank restricted SDP problem. No SDP solvers are used. For randomly generated instances,
Exploiting sparsity in linear and nonlinear matrix inequalities via positive semidefinite matrix completion
, 2010
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A highperformance software package for semidefinite programs: SDPA 7
, 2010
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(Robust) EdgeBased Semidefinite Programming Relaxation of Sensor Network Localization
 MATH PROGRAM
"... Recently Wang, Zheng, Boyd, and Ye (SIAM J Optim 19:655–673, 2008) proposed a further relaxation of the semidefinite programming (SDP) relaxation of the sensor network localization problem, named edgebased SDP (ESDP). In simulation, the ESDP is solved much faster by interiorpoint method than SDP r ..."
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Cited by 19 (2 self)
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Recently Wang, Zheng, Boyd, and Ye (SIAM J Optim 19:655–673, 2008) proposed a further relaxation of the semidefinite programming (SDP) relaxation of the sensor network localization problem, named edgebased SDP (ESDP). In simulation, the ESDP is solved much faster by interiorpoint method than SDP relaxation, and the solutions found are comparable or better in approximation accuracy. We study some key properties of the ESDP relaxation, showing that, when distances are exact, zero individual trace is not only sufficient, but also necessary for a sensor to be correctly positioned by an interior solution. We also show via an example that, when distances are inexact, zero individual trace is insufficient for a sensor to be accurately positioned by an interior solution. We then propose a noiseaware robust version of ESDP relaxation for which small individual trace is necessary and sufficient for a sensor to be accurately positioned by a certain analytic center solution, assuming the noise level is sufficiently small. For this analytic center solution, the position error for each sensor is shown to be in the order of the square root of its trace. Lastly, we propose a logbarrier penalty coordinate gradient descent method to find such an analytic center solution. In simulation, this method is much faster than interiorpoint method for solving ESDP, and the solutions found are comparable in approximation accuracy. Moreover, the method can distribute its computation over the sensors via local communication, making it practical for positioning and tracking in real time.
Universal Rigidity: Towards Accurate and Efficient Localization of Wireless Networks
, 2009
"... Abstract—A fundamental problem in wireless ad–hoc and sensor networks is that of determining the positions of nodes. Often, such a problem is complicated by the presence of nodes whose positions cannot be uniquely determined. Most existing work uses the notion of global rigidity from rigidity theory ..."
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Cited by 14 (4 self)
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Abstract—A fundamental problem in wireless ad–hoc and sensor networks is that of determining the positions of nodes. Often, such a problem is complicated by the presence of nodes whose positions cannot be uniquely determined. Most existing work uses the notion of global rigidity from rigidity theory to address the non–uniqueness issue. However, such a notion is not entirely satisfactory, as it has been shown that even if a network localization instance is known to be globally rigid, the problem of determining the node positions is still intractable in general. In this paper, we propose to use the notion of universal rigidity to bridge such disconnect. Although the notion of universal rigidity is more restrictive than that of global rigidity, it captures a large class of networks and is much more relevant to the efficient solvability of the network localization problem. Specifically, we show that both the problem of deciding whether a given network localization instance is universally rigid and the problem of determining the node positions of a universally rigid instance can be solved efficiently using semidefinite programming (SDP). Then, we give various constructions of universally rigid instances. In particular, we show that trilateration graphs are generically universally rigid, thus demonstrating not only the richness of the class of universally rigid instances, but also the fact that trilateration graphs possess much stronger geometric properties than previously known. Finally, we apply our results to design a novel edge sparsification heuristic that can reduce the size of the input network while provably preserving its original localization properties. One of the applications of such heuristic is to speed up existing convex optimization–based localization algorithms. Simulation results show that our speedup approach compares very favorably with existing ones, both in terms of accuracy and computation time.
Euclidean Distance Matrices and Applications
"... Over the past decade, Euclidean distance matrices, or EDMs, have been receiving increased attention for two main reasons. The first reason is that the many applications of EDMs, such as molecular conformation in bioinformatics, dimensionality reduction in machine learning and statistics, and especia ..."
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Cited by 14 (0 self)
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Over the past decade, Euclidean distance matrices, or EDMs, have been receiving increased attention for two main reasons. The first reason is that the many applications of EDMs, such as molecular conformation in bioinformatics, dimensionality reduction in machine learning and statistics, and especially
Beyond convex relaxation: A polynomial–time non–convex optimization approach to network localization
, 2013
"... AbstractThe successful deployment and operation of locationaware networks, which have recently found many applications, depends crucially on the accurate localization of the nodes. Currently, a powerful approach to localization is that of convex relaxation. In a typical application of this approa ..."
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Cited by 9 (2 self)
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AbstractThe successful deployment and operation of locationaware networks, which have recently found many applications, depends crucially on the accurate localization of the nodes. Currently, a powerful approach to localization is that of convex relaxation. In a typical application of this approach, the localization problem is first formulated as a rankconstrained semidefinite program (SDP), where the rank corresponds to the target dimension in which the nodes should be localized. Then, the nonconvex rank constraint is either dropped or replaced by a convex surrogate, thus resulting in a convex optimization problem. In this paper, we explore the use of a nonconvex surrogate of the rank function, namely the socalled Schatten quasinorm, in network localization. Although the resulting optimization problem is nonconvex, we show, for the first time, that a firstorder critical point can be approximated to arbitrary accuracy in polynomial time by an interiorpoint algorithm. Moreover, we show that such a firstorder point is already sufficient for recovering the node locations in the target dimension if the input instance satisfies certain established uniqueness properties in the literature. Finally, our simulation results show that in many cases, the proposed algorithm can achieve more accurate localization results than standard SDP relaxations of the problem.
Universal Rigidity and Edge Sparsification for Sensor Network Localization
, 2009
"... Owing to their high accuracy and ease of formulation, there has been great interest in applying convex optimization techniques, particularly that of semidefinite programming (SDP) relaxation, to tackle the sensor network localization problem in recent years. However, a drawback of such techniques is ..."
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Cited by 8 (1 self)
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Owing to their high accuracy and ease of formulation, there has been great interest in applying convex optimization techniques, particularly that of semidefinite programming (SDP) relaxation, to tackle the sensor network localization problem in recent years. However, a drawback of such techniques is that the resulting convex program is often expensive to solve. In order to speed up computation, various edge sparsification heuristics have been proposed, whose aim is to reduce the number of edges in the input graph. Although these heuristics do reduce the size of the convex program and hence making it faster to solve, they are often ad hoc in nature and do not preserve the localization properties of the input. As such, one often has to face a tradeoff between solution accuracy and computational effort. In this paper we propose a novel edge sparsification heuristic that can provably preserve the localization properties of the original input. At the heart of our heuristic is a graph decomposition procedure, which allows us to identify certain sparse generically universally rigid subgraphs of the input graph. Our computational results show that the proposed approach can significantly reduce the computational and memory complexities of SDP–based algorithms for solving the sensor network localization problem. Moreover, it compares favorably with existing speedup approaches, both in terms of accuracy and solution time. 1
Distributed Maximum Likelihood Sensor Network Localization
 IEEE Transactions on Signal Processing
, 2014
"... Abstract—We propose a class of convex relaxations to solve the sensor network localization problem, based on a maximum likelihood (ML) formulation. This class, as well as the tightness of the relaxations, depends on the noise probability density function (PDF) of the collected measurements.We deri ..."
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Cited by 7 (3 self)
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Abstract—We propose a class of convex relaxations to solve the sensor network localization problem, based on a maximum likelihood (ML) formulation. This class, as well as the tightness of the relaxations, depends on the noise probability density function (PDF) of the collected measurements.We derive a computational efficient edgebased version of this ML convex relaxation class and we design a distributed algorithm that enables the sensor nodes to solve these edgebased convex programs locally by communicating only with their close neighbors. This algorithm relies on the alternating direction method of multipliers (ADMM), it converges to the centralized solution, it can run asynchronously, and it is computation errorresilient. Finally, we compare our proposed distributed scheme with other available methods, both analytically and numerically, and we argue the added value of ADMM, especially for largescale networks. Index Terms—Distributed optimization, convex relaxations, sensor network localization, distributed algorithms, ADMM, distributed localization, sensor networks, maximum likelihood. I.